Question on Probability & Uniform Distribution.

[Sunil12]

Suppose a sample of random size N is taken from the continuous uniform(0, θ)
distribution, and N has a discrete distribution with p.m.f.

P (N = n) = 1/(n! (e − 1) ) for n = 1, 2, 3, . . . .

Determine the distribution of the
i) first order statistic (the minimum) of X1 , X2, . . . , XN .
ii) highest order statistic (the maximum) of X1, X2, . . ., XN .

Please help me to solve this problem.

[statdad]

Sounds like homework. What have you tried? What do you know in general about the distribution of order statistics?

[Sunil12]

Generally the distribution of Xi in a order statistic is Binomial. Isn't it ? because given a value 'a' Xi is either Xi < a or Xi>=a. It is like a success failure.

[bpet]

To get you started, P[max<=x|N=n] = P[X1<=x,...,Xn<=x]=P[X1<=x]^n, then simply take the expectation wrt N.

[statdad]

Generally the distribution of Xi in a order statistic is Binomial. Isn't it ? because given a value 'a' Xi is either Xi < a or Xi>=a. It is like a success failure.

Not quite. If you know that


P(X_n \le a)


( X_n is the largest order statistic) then you know that ALL the other values are less than or equal to a, so


P(X_n \le a) = P(X1 \le a \text{ and } X2 \le a \text{ and } \cdots \text{ and } Xn \le a)
= \left(P(X \le a)\right)^n = F(a)^n


by independence. To work with the minimum start with


P(X_{(1)} > a)


and think about what it means for the smallest item in the sample to be larger than some value.

[Sunil12]

FX1(a) = 1 - P(X1 > a)

which will essentially be 1 - (1 - Fx(x))n

Right ?